{"id":3085,"date":"2024-06-12T16:47:19","date_gmt":"2024-06-12T16:47:19","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=3085"},"modified":"2024-08-05T02:25:16","modified_gmt":"2024-08-05T02:25:16","slug":"limits-at-infinity-and-asymptotes-learn-it-5","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/limits-at-infinity-and-asymptotes-learn-it-5\/","title":{"raw":"Limits at Infinity and Asymptotes: Learn It 5","rendered":"Limits at Infinity and Asymptotes: Learn It 5"},"content":{"raw":"<h2>End Behavior Cont.<\/h2>\r\n<h3>Determining End Behavior for Transcendental Functions<\/h3>\r\n<h4>Trigonometric functions<\/h4>\r\n<p id=\"fs-id1165043327335\">The six basic trigonometric functions are periodic and do not approach a finite limit as [latex]x\\to \\pm \\infty[\/latex].<\/p>\r\n<p>For example, [latex]\\sin x[\/latex] oscillates between 1 and -1 (Figure 19). The tangent function [latex]x[\/latex] has an infinite number of vertical asymptotes as [latex]x\\to \\pm \\infty[\/latex]; therefore, it does not approach a finite limit nor does it approach [latex]\\pm \\infty [\/latex] as [latex]x\\to \\pm \\infty [\/latex] as shown in (Figure 20).<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211121\/CNX_Calc_Figure_04_06_011.jpg\" alt=\"The function f(x) = sin x is graphed.\" width=\"417\" height=\"197\" \/> Figure 19. The function [latex]f(x)= \\sin x[\/latex] oscillates between 1 and -1 as [latex]x\\to \\pm \\infty [\/latex][\/caption] [caption id=\"\" align=\"aligncenter\" width=\"500\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211123\/CNX_Calc_Figure_04_06_012.jpg\" alt=\"The function f(x) = tan x is graphed.\" width=\"500\" height=\"272\" \/> Figure 20. The function [latex]f(x)= \\tan x[\/latex] does not approach a limit and does not approach [latex]\\pm \\infty [\/latex] as [latex]x\\to \\pm \\infty [\/latex][\/caption]\r\n\r\n\r\n<h4>Exponential functions<\/h4>\r\n<p id=\"fs-id1165042459472\">Recall that for any base [latex]b&gt;0, \\, b\\ne 1[\/latex], the function [latex]y=b^x[\/latex] is an exponential function with domain [latex](\u2212\\infty ,\\infty )[\/latex] and range [latex](0,\\infty )[\/latex]. If [latex]b&gt;1, \\, y=b^x[\/latex] is increasing over [latex](\u2212\\infty ,\\infty )[\/latex].If [latex]0&lt;b&lt;1[\/latex], [latex]y=b^x[\/latex] is decreasing over [latex](\u2212\\infty ,\\infty )[\/latex].<\/p>\r\n<p>For the natural exponential function [latex]f(x)=e^x[\/latex], [latex]e\\approx 2.718&gt;1[\/latex]. Therefore, [latex]f(x)=e^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty )[\/latex] and the range is [latex](0,\\infty)[\/latex]. The exponential function [latex]f(x)=e^x[\/latex] approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty [\/latex] and approaches 0 as [latex]x\\to \u2212\\infty [\/latex].<\/p>\r\n<table id=\"fs-id1165042542966\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ex. After the header, the first row reads \u22125, \u22122, 0, 2, and 5. The second row reads 0.00674, 0.135, 1, 7.389, and 148.413.\">\r\n<caption>End behavior of the natural exponential function<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]-5[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]e^x[\/latex]<\/strong><\/td>\r\n<td>[latex]0.00674[\/latex]<\/td>\r\n<td>[latex]0.135[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]7.389[\/latex]<\/td>\r\n<td>[latex]148.413[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"267\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211126\/CNX_Calc_Figure_04_06_013.jpg\" alt=\"The function f(x) = ex is graphed.\" width=\"267\" height=\"234\" \/> Figure 21. The exponential function approaches zero as [latex]x\\to \u2212\\infty [\/latex] and approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].[\/caption]\r\n\r\n\r\n<p id=\"fs-id1165043218124\">Recall that the natural logarithm function [latex]f(x)=\\ln (x)[\/latex] is the inverse of the natural exponential function [latex]y=e^x[\/latex]. Therefore, the domain of [latex]f(x)=\\ln (x)[\/latex] is [latex](0,\\infty )[\/latex] and the range is [latex](\u2212\\infty ,\\infty )[\/latex].<\/p>\r\n<p>The graph of [latex]f(x)=\\ln (x)[\/latex] is the reflection of the graph of [latex]y=e^x[\/latex] about the line [latex]y=x[\/latex]. Therefore, [latex]\\ln (x)\\to \u2212\\infty [\/latex] as [latex]x\\to 0^+[\/latex] and [latex]\\ln (x)\\to \\infty [\/latex] as [latex]x\\to \\infty [\/latex].<\/p>\r\n<table id=\"fs-id1165042460463\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ln(x). After the header, the first row reads 0.01, 0.1, 1, 10, and 100. The second row reads \u22124.605, \u22122.303, 0, 2.303, and 4.605.\">\r\n<caption>End behavior of the natural logarithm function<\/caption>\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\r\n<td>[latex]0.01[\/latex]<\/td>\r\n<td>[latex]0.1[\/latex]<\/td>\r\n<td>[latex]1[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td><strong>[latex]\\ln (x)[\/latex]<\/strong><\/td>\r\n<td>[latex]-4.605[\/latex]<\/td>\r\n<td>[latex]-2.303[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<td>[latex]2.303[\/latex]<\/td>\r\n<td>[latex]4.605[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"417\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211128\/CNX_Calc_Figure_04_06_014.jpg\" alt=\"The function f(x) = ln(x) is graphed.\" width=\"417\" height=\"272\" \/> Figure 22. The natural logarithm function approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].[\/caption]\r\n\r\n\r\n<section class=\"textbox example\">\r\n<p>Find the limits as [latex]x\\to \\infty [\/latex] and [latex]x\\to \u2212\\infty [\/latex] for [latex]f(x)=\\frac{(2+3e^x)}{(7-5e^x)}[\/latex] and describe the end behavior of [latex]f[\/latex].<\/p>\r\n<p>[reveal-answer q=\"fs-id1165042711624\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1165042711624\"]<\/p>\r\n<p id=\"fs-id1165042711624\">To find the limit as [latex]x\\to \\infty[\/latex], divide the numerator and denominator by [latex]e^x[\/latex]:<\/p>\r\n<div id=\"fs-id1165042711652\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\underset{x\\to \\infty }{\\lim}f(x) &amp; =\\underset{x\\to \\infty }{\\lim}\\frac{2+3e^x}{7-5e^x} \\\\ &amp; =\\underset{x\\to \\infty }{\\lim}\\frac{(2\/e^x)+3}{(7\/e^x)-5}. \\end{array}[\/latex]<\/div>\r\n<p id=\"fs-id1165042499478\">As shown in Figure 21, [latex]e^x\\to \\infty [\/latex] as [latex]x\\to \\infty[\/latex]. Therefore,<\/p>\r\n<div id=\"fs-id1165042499514\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{2}{e^x}=0=\\underset{x\\to \\infty }{\\lim}\\frac{7}{e^x}[\/latex].<\/div>\r\n<p id=\"fs-id1165042499577\">We conclude that [latex]\\underset{x\\to \\infty }{\\lim}f(x)=-\\frac{3}{5}[\/latex], and the graph of [latex]f[\/latex] approaches the horizontal asymptote [latex]y=-\\frac{3}{5}[\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\r\n<p>To find the limit as [latex]x\\to \u2212\\infty[\/latex], use the fact that [latex]e^x \\to 0[\/latex] as [latex]x\\to \u2212\\infty [\/latex] to conclude that [latex]\\underset{x\\to -\\infty }{\\lim}f(x)=\\frac{2}{7}[\/latex], and therefore the graph of approaches the horizontal asymptote [latex]y=\\frac{2}{7}[\/latex] as [latex]x\\to \u2212\\infty[\/latex].<\/p>\r\n<p>Watch the following video to see the worked solution to this example.<\/p>\r\n<p><iframe title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=1453&amp;end=1547&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\r\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes1453to1547_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of \"4.6 Limits at Infinity and Asymptotes\" here (opens in new window)<\/a>.[\/hidden-answer]<\/p>\r\n<\/section>","rendered":"<h2>End Behavior Cont.<\/h2>\n<h3>Determining End Behavior for Transcendental Functions<\/h3>\n<h4>Trigonometric functions<\/h4>\n<p id=\"fs-id1165043327335\">The six basic trigonometric functions are periodic and do not approach a finite limit as [latex]x\\to \\pm \\infty[\/latex].<\/p>\n<p>For example, [latex]\\sin x[\/latex] oscillates between 1 and -1 (Figure 19). The tangent function [latex]x[\/latex] has an infinite number of vertical asymptotes as [latex]x\\to \\pm \\infty[\/latex]; therefore, it does not approach a finite limit nor does it approach [latex]\\pm \\infty[\/latex] as [latex]x\\to \\pm \\infty[\/latex] as shown in (Figure 20).<\/p>\n<figure style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211121\/CNX_Calc_Figure_04_06_011.jpg\" alt=\"The function f(x) = sin x is graphed.\" width=\"417\" height=\"197\" \/><figcaption class=\"wp-caption-text\">Figure 19. The function [latex]f(x)= \\sin x[\/latex] oscillates between 1 and -1 as [latex]x\\to \\pm \\infty [\/latex]<\/figcaption><\/figure>\n<figure style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211123\/CNX_Calc_Figure_04_06_012.jpg\" alt=\"The function f(x) = tan x is graphed.\" width=\"500\" height=\"272\" \/><figcaption class=\"wp-caption-text\">Figure 20. The function [latex]f(x)= \\tan x[\/latex] does not approach a limit and does not approach [latex]\\pm \\infty [\/latex] as [latex]x\\to \\pm \\infty [\/latex]<\/figcaption><\/figure>\n<h4>Exponential functions<\/h4>\n<p id=\"fs-id1165042459472\">Recall that for any base [latex]b>0, \\, b\\ne 1[\/latex], the function [latex]y=b^x[\/latex] is an exponential function with domain [latex](\u2212\\infty ,\\infty )[\/latex] and range [latex](0,\\infty )[\/latex]. If [latex]b>1, \\, y=b^x[\/latex] is increasing over [latex](\u2212\\infty ,\\infty )[\/latex].If [latex]0<b<1[\/latex], [latex]y=b^x[\/latex] is decreasing over [latex](\u2212\\infty ,\\infty )[\/latex].<\/p>\n<p>For the natural exponential function [latex]f(x)=e^x[\/latex], [latex]e\\approx 2.718>1[\/latex]. Therefore, [latex]f(x)=e^x[\/latex] is increasing on [latex](\u2212\\infty ,\\infty )[\/latex] and the range is [latex](0,\\infty)[\/latex]. The exponential function [latex]f(x)=e^x[\/latex] approaches [latex]\\infty[\/latex] as [latex]x\\to \\infty[\/latex] and approaches 0 as [latex]x\\to \u2212\\infty[\/latex].<\/p>\n<table id=\"fs-id1165042542966\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ex. After the header, the first row reads \u22125, \u22122, 0, 2, and 5. The second row reads 0.00674, 0.135, 1, 7.389, and 148.413.\">\n<caption>End behavior of the natural exponential function<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]-5[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]e^x[\/latex]<\/strong><\/td>\n<td>[latex]0.00674[\/latex]<\/td>\n<td>[latex]0.135[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]7.389[\/latex]<\/td>\n<td>[latex]148.413[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 267px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211126\/CNX_Calc_Figure_04_06_013.jpg\" alt=\"The function f(x) = ex is graphed.\" width=\"267\" height=\"234\" \/><figcaption class=\"wp-caption-text\">Figure 21. The exponential function approaches zero as [latex]x\\to \u2212\\infty [\/latex] and approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].<\/figcaption><\/figure>\n<p id=\"fs-id1165043218124\">Recall that the natural logarithm function [latex]f(x)=\\ln (x)[\/latex] is the inverse of the natural exponential function [latex]y=e^x[\/latex]. Therefore, the domain of [latex]f(x)=\\ln (x)[\/latex] is [latex](0,\\infty )[\/latex] and the range is [latex](\u2212\\infty ,\\infty )[\/latex].<\/p>\n<p>The graph of [latex]f(x)=\\ln (x)[\/latex] is the reflection of the graph of [latex]y=e^x[\/latex] about the line [latex]y=x[\/latex]. Therefore, [latex]\\ln (x)\\to \u2212\\infty[\/latex] as [latex]x\\to 0^+[\/latex] and [latex]\\ln (x)\\to \\infty[\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\n<table id=\"fs-id1165042460463\" class=\"column-header\" summary=\"The table has two rows and six columns. The first column is a header column and it reads x and ln(x). After the header, the first row reads 0.01, 0.1, 1, 10, and 100. The second row reads \u22124.605, \u22122.303, 0, 2.303, and 4.605.\">\n<caption>End behavior of the natural logarithm function<\/caption>\n<tbody>\n<tr valign=\"top\">\n<td><strong>[latex]x[\/latex]<\/strong><\/td>\n<td>[latex]0.01[\/latex]<\/td>\n<td>[latex]0.1[\/latex]<\/td>\n<td>[latex]1[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><strong>[latex]\\ln (x)[\/latex]<\/strong><\/td>\n<td>[latex]-4.605[\/latex]<\/td>\n<td>[latex]-2.303[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<td>[latex]2.303[\/latex]<\/td>\n<td>[latex]4.605[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure style=\"width: 417px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/2332\/2018\/01\/11211128\/CNX_Calc_Figure_04_06_014.jpg\" alt=\"The function f(x) = ln(x) is graphed.\" width=\"417\" height=\"272\" \/><figcaption class=\"wp-caption-text\">Figure 22. The natural logarithm function approaches [latex]\\infty [\/latex] as [latex]x\\to \\infty[\/latex].<\/figcaption><\/figure>\n<section class=\"textbox example\">\n<p>Find the limits as [latex]x\\to \\infty[\/latex] and [latex]x\\to \u2212\\infty[\/latex] for [latex]f(x)=\\frac{(2+3e^x)}{(7-5e^x)}[\/latex] and describe the end behavior of [latex]f[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1165042711624\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1165042711624\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165042711624\">To find the limit as [latex]x\\to \\infty[\/latex], divide the numerator and denominator by [latex]e^x[\/latex]:<\/p>\n<div id=\"fs-id1165042711652\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\begin{array}{ll} \\underset{x\\to \\infty }{\\lim}f(x) & =\\underset{x\\to \\infty }{\\lim}\\frac{2+3e^x}{7-5e^x} \\\\ & =\\underset{x\\to \\infty }{\\lim}\\frac{(2\/e^x)+3}{(7\/e^x)-5}. \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165042499478\">As shown in Figure 21, [latex]e^x\\to \\infty[\/latex] as [latex]x\\to \\infty[\/latex]. Therefore,<\/p>\n<div id=\"fs-id1165042499514\" class=\"equation unnumbered\" style=\"text-align: center;\">[latex]\\underset{x\\to \\infty }{\\lim}\\frac{2}{e^x}=0=\\underset{x\\to \\infty }{\\lim}\\frac{7}{e^x}[\/latex].<\/div>\n<p id=\"fs-id1165042499577\">We conclude that [latex]\\underset{x\\to \\infty }{\\lim}f(x)=-\\frac{3}{5}[\/latex], and the graph of [latex]f[\/latex] approaches the horizontal asymptote [latex]y=-\\frac{3}{5}[\/latex] as [latex]x\\to \\infty[\/latex].<\/p>\n<p>To find the limit as [latex]x\\to \u2212\\infty[\/latex], use the fact that [latex]e^x \\to 0[\/latex] as [latex]x\\to \u2212\\infty[\/latex] to conclude that [latex]\\underset{x\\to -\\infty }{\\lim}f(x)=\\frac{2}{7}[\/latex], and therefore the graph of approaches the horizontal asymptote [latex]y=\\frac{2}{7}[\/latex] as [latex]x\\to \u2212\\infty[\/latex].<\/p>\n<p>Watch the following video to see the worked solution to this example.<\/p>\n<p><iframe loading=\"lazy\" title=\"YouTube video player\" src=\"https:\/\/www.youtube.com\/embed\/0OVSQCWCzqc?controls=0&amp;start=1453&amp;end=1547&amp;autoplay=0\" width=\"750\" height=\"450\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<p>For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.<\/p>\n<p>You can view the <a href=\"https:\/\/oerfiles.s3.us-west-2.amazonaws.com\/Calculus\/Calculus1+Videos\/4.6LimitsAtInfinityAndAsymptotes1453to1547_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for this segmented clip of &#8220;4.6 Limits at Infinity and Asymptotes&#8221; here (opens in new window)<\/a>.<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":9,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":292,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3085"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3085\/revisions"}],"predecessor-version":[{"id":3115,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3085\/revisions\/3115"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/292"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/3085\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=3085"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=3085"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=3085"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=3085"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}